no. 3
On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms
[Conjecture de type BSD précisée pour des formes modulaires]
Ota, Kazuto1
1 Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1319-1364.

Sous certaines hypothèses, on prouve la partie rang de la conjecture précisée de Mazur–Tate de type BSD. Plus concrètement, on prouve que le rang du groupe de Selmer d’une forme modulaire elliptique est inférieur ou égal à l’ordre des zéros des éléments de Mazur–Tate, qui sont des éléments de certains algèbres de groupes construits à partir de valeurs spéciales de la fonction L associée. Notre résultat principal est considéré comme une généralisation de nos travaux antérieurs sur les courbes elliptiques.

Under some assumptions, we prove the rank-part of the Mazur–Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur–Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated L-function. Our main result is regarded as a generalization of our previous work on elliptic curves.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3557
Classification : 11G40, 11F11, 11R23
Keywords: Mazur–Tate refined conjectures, elliptic modular forms, $p$-adic $L$-functions
Mot clés : conjectures de Mazur-Tate précisée, formes modulaires elliptiques, fonctions $L$ $p$-adiques
Ota, Kazuto 1

1 Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2023__73_3_1319_0,
     author = {Ota, Kazuto},
     title = {On the rank-part of the {Mazur{\textendash}Tate} refined conjecture for higher weight modular forms},
     journal = {Annales de l'Institut Fourier},
     pages = {1319--1364},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {3},
     year = {2023},
     doi = {10.5802/aif.3557},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3557/}
}
TY  - JOUR
AU  - Ota, Kazuto
TI  - On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 1319
EP  - 1364
VL  - 73
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3557/
DO  - 10.5802/aif.3557
LA  - en
ID  - AIF_2023__73_3_1319_0
ER  - 
%0 Journal Article
%A Ota, Kazuto
%T On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms
%J Annales de l'Institut Fourier
%D 2023
%P 1319-1364
%V 73
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3557/
%R 10.5802/aif.3557
%G en
%F AIF_2023__73_3_1319_0
Ota, Kazuto. On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms. Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1319-1364. doi : 10.5802/aif.3557. https://aif.centre-mersenne.org/articles/10.5802/aif.3557/

[1] Amice, Yvette; Vélu, Jacques Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (Astérisque), Volume 24-25, Société Mathématique de France, 1975, pp. 119-131 | Numdam | MR | Zbl

[2] Benois, Denis On Iwasawa theory of crystalline representations, Duke Math. J., Volume 104 (2000) no. 2, pp. 211-267 | DOI | MR | Zbl

[3] Benois, Denis; Berger, Laurent Théorie d’Iwasawa des représentations cristallines. II, Comment. Math. Helv., Volume 83 (2008) no. 3, pp. 603-677 | DOI | MR | Zbl

[4] Berger, Laurent Bloch and Kato’s exponential map: three explicit formulas, Doc. Math., Volume Extra Vol. (2003), pp. 99-129 (Kazuya Kato’s fiftieth birthday) | MR | Zbl

[5] Berger, Laurent Limites de représentations cristallines, Compos. Math., Volume 140 (2004) no. 6, pp. 1473-1498 | DOI | MR | Zbl

[6] Berger, Laurent; Li, Hanfeng; Zhu, Hui June Construction of some families of 2-dimensional crystalline representations, Math. Ann., Volume 329 (2004) no. 2, pp. 365-377 | DOI | MR | Zbl

[7] Bloch, Spencer; Kato, Kazuya L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 333-400 | MR | Zbl

[8] Darmon, Henri A refined conjecture of Mazur-Tate type for Heegner points, Invent. Math., Volume 110 (1992) no. 1, pp. 123-146 | DOI | MR | Zbl

[9] Delbourgo, Daniel Elliptic curves and big Galois representations, London Mathematical Society Lecture Note Series, 356, Cambridge University Press, 2008, x+281 pages | DOI | MR

[10] Diamond, Fred; Shurman, Jerry A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005, xvi+436 pages | MR

[11] Dokchitser, Tim; Dokchitser, Vladimir On the Birch-Swinnerton–Dyer quotients modulo squares, Ann. Math., Volume 172 (2010) no. 1, pp. 567-596 | DOI | MR | Zbl

[12] Gomez, Clément On Mazur–Tate type conjectures for quadratic imaginary fields and elliptic curves, Ph. D. Thesis, McGill University (2014)

[13] Kato, Kazuya p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques. III (Astérisque), Volume 295, Société Mathématique de France, 2004, pp. 117-290 | Numdam | MR | Zbl

[14] Kim, Chan-Ho An anticyclotomic Mazur-Tate conjecture for modular forms, 2018 | arXiv

[15] Kobayashi, Shin-ichi Iwasawa theory for elliptic curves at supersingular primes, Invent. Math., Volume 152 (2003) no. 1, pp. 1-36 | DOI | MR | Zbl

[16] Kurihara, Masato On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I, Invent. Math., Volume 149 (2002) no. 1, pp. 195-224 | DOI | MR | Zbl

[17] Kurihara, Masato Refined Iwasawa theory for p-adic representations and the structure of Selmer groups, Münster J. Math., Volume 7 (2014) no. 1, pp. 149-223 | MR | Zbl

[18] Lei, Antonio Iwasawa theory for modular forms at supersingular primes, Compos. Math., Volume 147 (2011) no. 3, pp. 803-838 | DOI | MR | Zbl

[19] Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia Euler systems for Rankin-Selberg convolutions of modular forms, Ann. Math., Volume 180 (2014) no. 2, pp. 653-771 | DOI | MR | Zbl

[20] Longo, Matteo; Vigni, Stefano A refined Beilinson-Bloch conjecture for motives of modular forms, Trans. Am. Math. Soc., Volume 369 (2017) no. 10, pp. 7301-7342 | DOI | MR | Zbl

[21] Mazur, Barry; Rubin, Karl Kolyvagin systems, Mem. Am. Math. Soc., Volume 168 (2004) no. 799, p. viii+96 | DOI | MR | Zbl

[22] Mazur, Barry; Tate, John Refined conjectures of the “Birch and Swinnerton–Dyer type”, Duke Math. J., Volume 54 (1987) no. 2, pp. 711-750 | DOI | MR | Zbl

[23] Mazur, Barry; Tate, John; Teitelbaum, Jeremy On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48 | DOI | MR | Zbl

[24] Nekovář, Jan Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math., Volume 107 (1992) no. 1, pp. 99-125 | DOI | MR | Zbl

[25] Nekovář, Jan Selmer complexes, Astérisque, 310, Société Mathématique de France, 2006, viii+559 pages | MR | Zbl

[26] Ota, Kazuto Kato’s Euler system and the Mazur-Tate refined conjecture of BSD type, Am. J. Math., Volume 140 (2018) no. 2, pp. 495-542 | DOI | MR | Zbl

[27] Otsuki, Rei Construction of a homomorphism concerning Euler systems for an elliptic curve, Tokyo J. Math., Volume 32 (2009) no. 1, pp. 253-278 | DOI | MR | Zbl

[28] Perrin-Riou, Bernadette Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math., Volume 115 (1994) no. 1, pp. 81-161 (with an appendix by Jean-Marc Fontaine) | DOI | MR | Zbl

[29] Perrin-Riou, Bernadette Théorie d’Iwasawa et loi explicite de réciprocité, Doc. Math., Volume 4 (1999), pp. 219-273 | MR | Zbl

[30] Ribet, Kenneth A. On l-adic representations attached to modular forms. II, Glasg. Math. J., Volume 27 (1985), pp. 185-194 | DOI | MR | Zbl

[31] Rubin, Karl Euler systems (Hermann Weyl Lectures), Annals of Mathematics Studies, 147, Princeton University Press, 2000, xii+227 pages | DOI | MR

[32] Serre, Jean-Pierre; Tate, John Good reduction of abelian varieties, Ann. Math., Volume 88 (1968), pp. 492-517 | DOI | MR | Zbl

[33] Tilouine, Jacques Modular forms and Galois representations, Bull. Greek Math. Soc., Volume 46 (2002), pp. 63-78 | MR | Zbl

[34] Višik, M. M. Nonarchimedean measures associated with Dirichlet series, Mat. Sb., N. Ser., Volume 99(141) (1976) no. 2, p. 248-260, 296 | MR

Cité par Sources :